Piece rates with endogenous monitoring : some theory and evidence

EUROPEAN UNIVERSITY INSTITUTE, FLORENCE

DEPARTMENT O F ECONOMICS

EU I W O R K I N G P AP E R No. 88/335

PIECE RATES WITH

iENOUS MONITORING:

• m e T heory and Evidence

£

hy

. FITZROY * and Kornelius KRAFT '

* Science Centre, Berlin

** Kassel University, F. R. G.

This paper was written while the first author was a Jean Monnet Fellow at the European

University Institute. We are grateful to Harris Schlesinger and Joseph Stiglitz for valuable

discussions.

BADIA FIESOLANA, SAN DOMENICO (F I)

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No part of this paper may be reproduced in any form without

permission of the author.

(C) Felix FitzRoy and Kornelius Kraft Printed in Italy in March 1988

European University Institute Badia Fiesolana 50016 San Domenico (Fi)

-Italy

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Piece Rates with Endogenous Monitoring:

Some Theory and Evidence

Felix R. FitzRoy and Kornelius Kraft

* European University Institute, Florence, ana Science Centre, 3erlin ** Kassel University, F.R.G.

Ab stract

A model of piece rates under uncertainty with endogenous mon­ itoring which yields a realistic combination of positive time and piece rates in competitive equilibrium is developed, contrasting with customary models yielding a negative time rate unless an ar­ bitrary constraint is imposed. Piece rates and earnings need not be monotonicallv related to exogenous skill or task complexity in equilibrium. Empirical tests confirmed non-monoticity, but sur­ prisingly found very weak overall effects of piece rates. Greater conflict potential of piece pay is supported by a positive rela­ tionship with unionization and probability of a works council.

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1. Introduction

Most payment systems in firms involve a mixture of perform­

ance-related rewards or incentive pay, and a basic wage or time-

rate which depends primarily on the job classification and on

personal characteristics such as seniority. Microanalysis of in­

centive and welfare effects is clearly important to understanding

both the behaviour of the firm and of labour markets in the ag­

gregate, yet there has been remarkably little empirical testing of

standard assumptions and theories with appropriate microdata. In

contrast to this neglect, there has been considerable theoretical

progress in the understanding of incentives in the firm, starting

with the stimulating contribution of Stiglitz (1975), who argued

that risk aversion and the difficulty of monitoring true output

(including quality) should reduce piece rates relative to time

rates. This suggests that monitoring itself should be treated as

an endogenous variable which is determined by the firm as part of

an optimal contract. While recent writers such as Eaton and White

(1984) and Ordover and Shapiro (1984) do model endogenous monitor­

ing, a problem with their models discussed further below is ' that

they have to restrict pay to the empirically rare case of a pure

piece rate.

Discontinuous incentives (prizes and penalties) are also

important in practice, and have been studied by Lazear and Rosen

(1981), Nalebuff and Stiglitz (1983) and others under the assump­

tion that monitoring and technology are exogenously given. In the

absence of piece rates there is presumably always some minimal

performance standard which job-holders have to meet, and there may

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also be 'prizes' in the form of promotion possibilities even when

explicit bonuses are not paid. Monitoring in some form or other is

thus always necessary, though continuous quantification of per­

formance for piece pay is likely to require more intensive monit­

oring in most circumstances.

Monitoring cost can plausibly be reduced through the long­

term association of workers with specific skills (FitzRoy and

Mueller, 1984; Williamson, 1985). On the other hand, machinepaced

work involving repetitive tasks and little skill can also econom­

ize on the costs of supervision (Edwards, 1979). Piece rates may

then be irrelevant, and the simple negative relationship between

use of piece rates and skill or task complexity does not necessar­

ily hold.

In this paper we first criticize the usual models of endogen­

ous monitoring for their failure to generate an interior solution

with a positive time rate as well as a piece rate. We then develop

an alternative model without this drawback which also yields some

testable implications (section 2). In section 3 we describe a data

set containing information on piece rates and proxies for skill.

OLS and Tobit regression results are presented in section 4, and a

summary conclusion follows.

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2. Models of Incentives with Endogenous Monitoring

A. Critique of the standard model

The standard model of incentive pay under uncertainty

assumes there is some probability, say P > 0 , of observing true

output or effort, e £-0, which can then be rewarded with a piece

rate, d >0 , and a basic (time) wage, w, to yield income

y

0

When effort is unobserved with probability 1-P, a wage-income =

z >0 is paid. Assume for simplicity that the worker has a well 2

behaved, separable utility function given by

U(e, y) = u(y) - D(e), (2)

so that expected utility when the worker chooses effort = e is

V = Pu(yQ ) + (1-P) u(z) - D(e). (3)

The worker then chooses optimal effort e to satisfy the first-

order condition on (3),

Ppu'(yQ ) = D'(e), (4)

where y^ = pe+w, and optimal e = e(P,p,w) for all positive P <1.

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Next, we assume constant returns to workers and a production

function of effort

q = q(e), (5)

which is concave increasing. Neglecting monitoring costs for the

moment, the employer is then assumed to maximize profit per worker

by choice of pay parameter, while holding expected utility con­

stant. The Lagrangian for this problem can be written

L = q(e) - PyQ - (1-P) z + u(V-A), (6)

where A is alternative utility. Writing down the first-order

conditions with respect to P, p, w and z and setting

z _{= y '} (7)

say, yields the usual efficiency condition of equality between

marginal product and marginal rate of substitution, or

q'(e) = D'(e) / u'(y) . (8)

But comparison with (4) shows that optimal parameters then satisfy

Pp = q'(e ), ( 9 )

and the model can be closed by requiring zero expected profit in

competitive equilibrium.

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A surprising and somewhat disturbing feature of this model is

that, in spite of uncertainty, the solution (7)— (8) is in fact

first-best; there is no risk for workers, who always provide

optimal effort. Indeed, this can hold for any positive probability

P, for under appropriate assumptions, (9) can then be solved for

p = p(P,w), say, and w can then be chosen to satisfy the zero-

profit condition.3 The catch is that the wage w may then be nega­

tive, and to avoid this unrealistic case, the fixed wage is usual­

ly constrained to be non-negative.

Matters become even more unsatisfactory when monitoring costs

are introduced. Suppose there is a convex increasing cost M(P)

with M(0)=0 for observing true effort with probability P. Then

without a lower bound on the wage (w) there is no interior solu­

tion in general. Lowering P always reduces monitoring cost other­

wise for any P >0, but of course the solution breaks down with no

observation, or P=0. In this case the non-negativity constraint on

w leads to a solution with optimal w=0, and payment by a pure

piece rate, which is also rather unrealistic.

The problem with monitoring cost can be illustrated most

simply with the special case of risk-neutral workers. For effi­

cient contracts, competitive firms will maximize expected utility

subject to zero expected profits, so the Lagrangian is now

L = P(ep+w) + (1-P) z - D(e) + y[q(e) - PyQ - (1-P) z]- M. (10)

Clearly, Lz = 0 implies n=l (denoting partials by subcripts), and

the basic efficiency condition, which is now

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D'(e) = q ‘(e ) (1 1)

follows from L and L =0. But then

w p

L,

P (-D'+q’)ep - M'(P) M' < 0 (1 2)

so there is no interior solution with P >0 when marginal monitor­

ing cost is positive for all positive P. Of course, as P + 0, it

essentially follows from (4) that p °°, so in general the fixed

wage w -► - ® . As remarked above, it is precisely to avoid this

quite unrealistic case that an ad hoc constraint such as non-4

negativity is usually imposed on w.

B . An alternative model

In view of these problems we shall use a different model

here, where there is always uncertainty in observing individual

effort, but this uncertainty can be reduced by (costly) monitor­

ing. Firms are assumed to produce differentiated products, and

average revenue per worker is a known function of true effort, e,

and also of a parameter, x. This parameter represents average

specific training expenditure per worker necessary for production

in each particular firm, and is thus a proxy for the average skill

level required to make each product, or for average task complex­

ity. The firm's choice of product, and hence of training expendit­

ure, x, will be taken as historically and technologically given

for the present analysis,5 where we focus on optimal choice of

piece rates and monitoring.

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To obtain results it seems necessary to use some simple

functional forms as follows. A firm with a fixed number of workers

is assumed to have revenue per worker

R = ef(x ), (13)

where f(x) is concave increasing. Next, observed effort per worker

is a random variable

“e = e + e , (14)

where e is true effort, and the error e (m, x) is a decreasing

function of monitoring, m, and an increasing function of task

complexity, x. We assume that observation is unbiased, so that

E e = 0, (15)

where E is the expectation over a known distribution. It follows

that if workers are paid a fixed wage or time rate, w, and a piece

rate, p, per unit of observed effort, then income per unit time is

y = pe + w, (16)

and expected income is

Ey pe + w. (17)

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To proceed we shall assume identical workers with mean-

variance utility, and the error-variance (for computational simp­

licity)

2

x /2m. (18)

A worker's utility can then be written as follows to yield simple

solutions:

U = pe + w - [a2 (y-Ey)2 + e2 ]/2, (19)

yielding expected utility

V = EU = pe + w - [(apx)2/2m] - e2/2, (20)

where a >0 represents attitude to risk and is monotonically relat­

ed to Arrow-Pratt risk aversion. The worker's optimal effort, e,

is then simply

e =

(21)

by the first-order condition on (20).

Subject to the incentive-compatibility condition (21), the

firm is now assumed to maximize profit per worker holding expected

utility constant. The Lagrangian is

L it + u(V-VQ ), (2 2)

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where expected profit is revenue less total wage and non-wage

labour cost, and monitoring cost is m/2 for convenience,

tt = R ( e , x ) - x - p e - w - m/2. (23)

In equilibrium with free entry the time rate should be chosen so

that it=0, and expected income per worker is then

y = R ( e , x ) -x-m/2 = pe+w-x = p -t-w-x. (24)

An optimal contract or parameters (m, p), which are function of x,

can then be obtained by maximizing zero-profit expected utility,

which is

~ ~ 2 "2

V = R(e,x)-x-(m/2) - [(apx) /2m] - e / 2 . (25)

The first-order condition Vm=0 yields optimal monitoring per

worker

m = apx, (26)

where p is the optimal piece rate. Note that corner solutions with

either p=0 or ® cannot generate positive utility, so thatif an

interior maximum with positive utility exists, the other necessary

condition V =0 must also be satisfied. From (26), (21) and (13) P

this condition gives

= f(x) -P ax (27)

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for all (a, x) such that p > 0.

Then if f'(0) >a, p is an initially increasing function of x, but

p may subsequently decline. Thus there is no particular reason for

a generally negative monotonic relationship, as is often assumed.

Now it is straightforward to calculate equilibrium earnings

y = f(f-ax) - x - ax(f-ax)/2

= [(f-ax) (2f-ax)/2] - x, (28)

and the fixed wage is

W = y - p2' = x [a(f-ax) - 2]/2. (29)

Clearly, y is likely to be non-monotonic in general, and if f>ax,

then for large enough a, w will be positive.

This model thus generates some non-trivial and testable pre­

dictions. In particular, an interior unconstrained positive wage

is possible by (29), and w increases with risk aversion while

earnings decline. Finally we have the requirement or restriction

on parameters that unobservable utility should be positive in

equilibrium, since unemployment would be preferable otherwise. We

find that V [(f-ax)2/ 2 ] - x. (30)

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Clearly, this model offers no insights into the reasons why some firms have no piece rates. Employers can also motivate effort by offering promotion prospects in the long run, as well as by es­ tablishing work norms or production quotas as a condition of em­ ployment, as mentioned above. Homogeneity and high skill levels of the workforce seem likely to favour the absence of piece rates as well as the long-term attachment which in turn encourages specific training and career patterns. We return to this problem in the em­ pirical section 3 below.

Finally, a development with some theoretical appeal would be

to allow workers with differing risk aversion to choose their opt­

imal employer as defined by the skill parameter, x. However, other

personal attributes such as ability are also relevant, and the

model quickly becomes intractable. For our empirical purposes we

assume constant risk aversion (and ability), and note that earn­

ings differences across firms in our sample are extremely small.

The parameter x is not easily observable to potential employees

searching for a job, so it seems reasonable for our purposes to

assume that homogeneous workers choose jobs on the basis of more

obvious characteristics such as location, or availability.

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3. Empirical Evidence and Specification

While our available data are too limited to allow a complete

test of the model (which is of course too simple to capture all

relevant influences), we can explore some hypotheses of interest

in an area where empirical studies since Pencavel (1977) are rare.

In one of the most recent attempts, Seiler (1984) argues that

piece rates should be associated with greater variation of earn­

ings. He assumes as selfmevident that incentive pay should in­

crease effort, but does not consider the role of technology in

relation to monitqring and machine-paced work with time rates, nor

does he develop a formal model of incentives and other factors.

Comparing firms with and without piece-rate payment schemes, Sei­

ler finds, as expected, that earnings are greater and more dis­

persed in the former. Seiler argues that the incentive premium is

too large to be explained entirely as risk compensation, and con­

cludes that a substantial effort-incentive effect has been con­

firmed. Most of the workers in Seiler's sample were on an incent­

ive pay scheme, suggesting generally favourable technology and job

design. An additional problem is that "most human capital variab­

les are absent" (Seiler, 1984, p. 369), so it is not entirely

clear how these results should be interpreted.

The basis for our study is data collected in 65 medium-sized

firms in the German metal-working industry. Our unit of observa­

tion is hence not individual workers but firms. Of the 65 firms 62

provided data for 1979 and 61 for 1977. 60 percent of the firms

had no individual incentive pay, and in the others, piece payment

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ranged up to 100 percent of total pay. Although the sample could

be assigned to 3 classifications of the metal-working industry,

output was heterogeneous, with each firm essentially making (a)

different product(s). The data (collected in personal interviews)

included proxies for human capital, technology, and market power,

and are described in Table 1.

As dependent variables we use annual earnings (piece pay plus

time pay), and annual incentive payment.^ Our theory does not

predict an unambiguous- relationship between these variables and

complexity, and the latter is of course not directly observable.

As a proxy we use expenditure on training in the firm (TRAIN) with

a quadratic term to capture any non-linearity. The proportion of

output exported may be an additional proxy for skill or complexity

under the assumption that international market competition favours

higher quality products.

The collective "voice" of labour is represented by the Works

Council which employees have the right to elect, and which has

far-reaching powers under German labour legislation.7 Firms with

a (non-mandatory) Works Council are indicated by a dummy (WOCO).

UNION density is a related variable which presumably strengthens

the voice of labour in internal negotiations and disputes. Note

however that wage bargaining is centralized, though individual

employers may offer premiums when the labour market is tight.

Finally, some conventional explanatory variables are dummies

for sub-classifications of the metal industry, a time dummy for

1979, and urban-location dummy, and number employed in the firm

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(NEM). The capital ownership share held by top management (CTOP)

was also utilized.

Here we shall estimate a wage equation for firms with posit­

ive incentive payments (49 observations). Since our variables are

unlikely to capture all the determinants of piece rates, a sample-

selection bias can arise with OLS regressions in the subsample

with positive incentive pay.** To correct for this bias we follow

Heckman (1979), and estimate the probability of using piece rates

with a Probit regression. The density and c.d.f. are used to con­

struct the inverse Mills ratio, which is then added to the wage

equation as a correction variable.

In our second regression we attempt to explain the magnitude 9

of incentive pay, using Tobit regression for the whole sample.

The best results were obtained without industry or time dummies as

reported in Table 2.

4. Results

Incentive pay is positively related to annual wages in our

sample in accordance with theory. In the second equation, for

piece rates, the proxies for skill or complexity are negative and

significant (TRAIN, EXP) while these variables are positively rel-2

ated to earnings. However, the quadratic term (TRAIN ) is negative

in the wage equation. A possible explanation might be that small

amounts of training are basic and general, while further training

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tends to be specific and thus has no positive effect on earnings.

Overall/ these results are consistent with our model.

The Heckman correction is negative significant, indicating

selection bias as expected. Surprisingly, the urban dummy and

number employed were both negative in the wage equation. The Works

Council also has a negative effect, as we have discussed else-

10

where. An interesting finding is that WOCO and UNION are strong­

ly related to incentive pay, presumably because these institutions

of collective voice reduce the transactions costs of setting piece

rates. Alternatively, conflict over rate setting could encourage

unionization and the formation of a Works Council, and causality

might be reversed. Unfortunately, simultaneous estimation is in­

feasible with our data set.

Another noteworthy finding (in unreported regressions) was

that the hourly wage (rate) is not affected by incentive pay in

the subsample, so the influence on earnings must be via annual

hours worked. Clearly these are part of the total employment con­

tract which includes stochastic overtime with a frequency depend­

ing on demand and employment levels as well as "labour hoarding".

As one expects from identical mean annual earnings in the two

subsamples (with and without piece rates), an equation for annual

earnings estimated for the whole sample showed no influence of

incentive pay. The effect of piece rates in Table 2 is small and

only just significant, and overall we can conclude that the

results of piece rates are surprisingly weak, in contrast to

previous work as well as historical evidence (Clark, 1984).

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5. Conclus ions

A simple exploratory model of uncertainty and incentive pay

which does not -unrealistically- imply negative or constrained

zero time rates has yielded several surprising and even anomalous

results. One of these, a non-monotonic relationship between skill

or task complexity and total earnings, also emerged in our empir­

ical study. Data limitations precluded a full account, but the

uniformity of total earnings under maximum variation of piece

rates (from 0 to 100 percent) remains puzzling. In view of the

importance of incentives and agency problems in economic theory

and practice, there is clearly much scope for testing models with

improved microdata.

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Table 1

Mean value Mean value Definition Abbreviat ion in firms with in firms without

piece rates piece rates

Subsector dummy (ERM) ID 2 0.24 0.24 Subsector dummy (Machinery) ID 3 0.59 0.70

Dummy for existence of works council WOCO 0.96 0.69 Training expenditure per employee (1000 DM) TRAIN 0.69 0.97

Dummy for 1977 TIME 0.49 0.50

Percentage of work­ force unionized

UNION 52.35 27.5

Share of production EXP 29.74 30.6

Dummy for urban locat ion

URB 0.61 0.43

Number of employees NEM 999 385

Dummy = 1

when top management holds at least 25% of capital

CTOP 0.37 0.82

Yearly wages per worker (1000 DM)

WAGES 25.204 25.649

Incentive payments per worker and year (1000 DM) INCENTIVES 12.42 0

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Table 2

Independent Dependent variables

variables Wages Incentives

(Tobit) Constant 3.,73 -0.63 (13. 12) (-6.35) ID 2 -0.,07 (-0.,76) ID 3 -0.,02 (-o. 19) WOCO -0.,38 1.54 (-2..05) (3.12) TRAIN 0..28 -0.25 (3..04) (-1.69) (TRAIN)2 -0..13 (-3..60) TIME -0..04 (-0,.88) INCENTIVES 0,.09 (1 .94) UNION -0,.001 0.01 (-0,.66) (2.22) EXP 0..003 -0.03 (1..93) (-4.75) URB -0,.20 0.63 (-2,.74) (2.29) NEM -0..0001 0.003 (-3 .31) (2.76) CORRECTION -0 .67 (“2 .09) CTOP -1.01 (-3.22) R2 0 .58 n 49 123

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Footnotes

1. See e.g. Calvo and Wellisz (1979), Eaton and White (1984), and Ordover and Shapiro (1984), who include endogenous monit­ oring as discussed below.

2. That is, u is concave and D is convex (increasing).

3. For details, see FitzRoy (1985).

4. See the references cited in fn. 1 above.

5. Endogenous x was considered in previous versions, but led to inconclusive results.

6. Hourly earnings for the whole sample estimated in FitzRoy and Kraft (1985) were unaffected by incentive pay. See also re­ marks at the end of this section.

7. For a description of the German system of industrial rela­ tions and the role of the Works Council, see FitzRoy and Kraft (1985, 1986).

8. This point is not discussed by Seiler (1984), who also neg­ lects human capital. Thus, for example, if workers on piece rates were less skilled on average, but also paid a risk premium, these opposing effects might cancel out.

9. This is the first attempt of this kind which we are aware of.

10. In FitzRoy and Kraft (1985, 1986) 'we argue that the most ef­ ficient employers can offer higher wages and working condi­ tions which render the Works Council superfluous. We also find a negative relation between the Works Council and factor productivity, and can thus reject the "efficiency voice theo­ ry" of Freeman and Medoff (1984) for our sample, as well as

on theoretical ground.

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References

Calvo, G.A. and Wellisz, S., "Hierarchy, Ability and Income Dis­ tribution", Journal of Political Economy 87 (October 1979), 991-1010.

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Williamson, 0. York: Free

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François DUCHENE Beyond the First C.A.P.

Domenico Mario NUTI Political and Economic Fluctuations in the Socialist System

Christophe DEISSENBERG On the Determination of Macroeconomic Policies with Robust Outcome

Domenico Mario NUTI A Critique of Orwell's Oligarchic Collectivism as an Economic System Will BARTLETT Optimal Employment and Investment

Policies in Self-Financed Producer Cooperatives

Jean JASKOLD GABSZEWICZ Asymmetric International Trade Paolo GARELLA

Jean JASKOLD GABSZEWICZ Subjective Price Search and Price

Paolo GARELLA Compétition

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Kumaraswamy VELUPILLAI Dwight M. JAFFEE

Gerd WEINRICH

Domenico Mario NUTI

Will BARTLETT

Will BARTLETT Gerd WEINRICH

Jesper JESPERSEN

Jean JASKOLD GA3SZEWICZ Paolo GARELLA

Domenico îlario NUTI Pierre DEHEZ Jean-Paul FITOUSSI

On Rationalizing Expectations

Term Structure Intermediation by Depository Institutions

Price and Wage Dynamics in a Simple Macroeconomic Model with Stochastic Rationing

Economic Planning in Market Economies: Scope, Instruments, Institutions Enterprise Investment and Public Consumption in a Self-Managed Economy Instability and Indexation in a Labour- Managed Economy - A General Equilibrium Quantity Rationing Approach

Some Reflexions on the Longer Term Con­ sequences of a Mounting Public Debt Scattered Sellers and Ill-Informed Buye A/Model of Price Dispersion

The Share Economy: Plausibility and Viability of Weitzman's Model Wage Indexation and Macroeconomic Fluctuations

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Derek C. JONES

Are There Life Cycles in Labor-Managed Firms? Evidence for France

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Competitive Equilibria with Increasing Returns

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Ouantity Guided Price Setting

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Equilibrium and the Role of the Firm in Incomplete Markets

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Income Distributions and the Axiom of Revealed Preference

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Exchange Rate Uncertainty and Foreign Trade

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Segmented Trends and Nonstationary Time Series

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Piece Rates with Endogenous Monitoring: Some theory and evidence

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87/284: Saul ESTRIN and Derek JONES

The Determinants of Workers' Participation and Productivity in Producer Cooperatives

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Pathways to Drug Abuse Amongst Girls in Britain and Holland

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Aspects of School Culture and the Social Control of Girls

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Becoming a Teenage Prostitute in Spain and the U.S.A.

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A comparison of crime and its

treatment amongst girls in Britain and Belgium

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Towards an effective policy for delinquent girls

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Computing, Economic Equilibria by Variable Dimension Algorithms: State of the Art

87/306: Paolo C. GARELLA Adverse Selection and Intermediation

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Income Distributions and the Axiom of Revealed Preference

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Exchange Rate Uncertainty and Foreign Trade

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87/315: Serge NOIRET Nuovi motivi per studiare i meccanismi delle leggi elettorali. Una

riflessione metodologica a proposito della legge del 1919 in Italia

87/316: Alain GOUSSOT Les sources internationales de la culture socialiste italienne à la fin du 19e siècle et au début du 20e siècle. Problèmes de la composition de l'idéologie du PSI et ses rapports avec la circulation des idées en Europe

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The Intrinsic Limits of Modern Economic Theory

An Equivalence Theorem for a Bargaining Set

"La classe la plus nombreuse, la plus utile et la plus précieuse".

Organizzazione del lavoro e conflitti nella Parigi rivoluzionaria.

Qu'est-ce que c'est "Le Patronat"? Quelques enjeux théoriques et observations empiriques

Decision-Making Processes, Conflicts, and Cabinet Government

88/328: Ida KOPPEN

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The European Community's Environment Policy.

From the Summit in Paris, 1972, to the Single European Act, 1987

Assessing Structural Change: The Case

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88/331: David CANNING Convergence to Equilibrium in a Sequence of Games with Learning

88/332: Dalia MARIN Trade and Scale Economies. A causality test for the U.S., Japan, Germany and the UK

88/333: Keith PILBEAM Fixed versus Floating Exchange Rates Revisited

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Die EWG und die Versalzung des Rheins

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Piece Rates with Endogenous Monitoring Some Theory and Evidence

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Bundesrepublik Deutschland, Frankreichs und Italiens

-r W o -r k i n g P a p e -r o u t o f p -r i n t

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